Most builders will want to see drawings before quoting. You can get ‘design and build’ firms that offer both, though I’ve found them to be more expensive overall.
If you know what you want to do with your extension, an architectural designer might be your best bet - they should be cheaper than an architect. But if you’re not sure, a good architect can help you make the most of the space. Either way, go with a personal recommendation if you can.
If not, if you know of houses locally that have been extended, look in the Planning Portal - if they’ve gone through planning permission, you can take a look at their drawings and see who designed them.
Once you have drawings, you can go out to builders to quote. That in itself (in my recent experience) can be a headache as they’re all so busy. We’ve contacted over 20 builders over the past year and had actual quotes from 6 of them. It does seem to be improving lately though. Lead times seem to have shortened a bit too - last year some were booking jobs 12-18 months ahead. Now it seems to be more like 6-12.
Prices are a lot higher than they were pre-Brexit but it’s definitely worth getting multiple quotes - our cheapest quote is just about half the price of our most expensive.
Speaking of costs, you need to factor in the price of everything that your builder won’t cover, eg flooring, lighting, kitchen cabinets, appliances, sockets, switches (if you want fancy ones 😊). Excel is your friend! For our build, it’s looking like an almost 50:50 split between the build and everything on top. But that will vary depending on your specs.
Hope that helps!